• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.293-309
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• 2011

The purpose of this article is to prove strong convergence theorems for weak relatively nonexpansive mapping which is firstly presented in this article. In order to get the strong convergence theorems for weak relatively nonexpansive mapping, the monotone CQ iteration method is presented and is used to approximate the fixed point of weak relatively nonexpansive mapping, therefore this article apply above results to prove the strong convergence theorems of zero point for maximal monotone operators in Banach spaces. Noting that, the CQ iteration method can be used for relatively nonexpansive mapping but it can not be used for weak relatively nonexpansive mapping. However, the monotone CQ method can be used for weak relatively nonexpansive mapping. The results of this paper modify and improve the results of S.Matsushita and W.Takahashi, and some others.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.87-102
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• 2011

The purpose of this paper is to prove strong convergence theorems for finding a common element of the set of fixed points of a weak relatively nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping by a new hybrid method in a Banach space. We shall give an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping in Banach space $l^2$. Our results improve and extend the corresponding results announced by Ying Liu[Ying Liu, Strong convergence theorem for relatively nonexpansive mapping and inverse-strongly-monotone mapping in a Banach space, Appl. Math. Mech. -Engl. Ed. 30(7)(2009), 925-932] and some others.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1453-1462
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• 2011

In this paper, we study a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By using the KKM technique and the concept of the Hausdorff metric, we obtain some existence results for generalized nonlinear variational-like inequalities with generalized monotone multi-valued mappings in Banach spaces. These results improve and generalize many known results in recent literature.

• East Asian mathematical journal
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• v.29 no.3
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• pp.315-326
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• 2013

In this paper, a new monotonicity, $M({\cdot},{\cdot})$-monotonicity, is introduced in Banach spaces, and the resolvent operator of an $M({\cdot},{\cdot})$-monotone operator is proved to be single valued and Lipschitz continuous. By using the resolvent operator technique associated with $M({\cdot},{\cdot})$-monotone operators, we construct a proximal point algorithm for solving a class of variational inclusions. And we prove the convergence of the sequences generated by the proximal point algorithms in Banach spaces. The results in this paper extend and improve some known results in the literature.

• Nonlinear Functional Analysis and Applications
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• v.29 no.1
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• pp.15-33
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• 2024

In this article, we considered a class of nonlinear variational hemivariational inequality problems and investigated a gap function and regularized gap function for the problems. We discussed the global error bounds for such inequalities in terms of gap function and regularized gap functions by utilizing the Clarke generalized gradient, relaxed monotonicity, and relaxed Lipschitz continuous mappings. Finally, as applications, we addressed an application to non-stationary non-smooth semi-permeability problems.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.257-265
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• 2011

In this paper, we first prove a common fixed point theorem for generalized nonlinear contraction mappings in complete metric spaces there by generalizing and extending some known results. Then we present this result in the context of ordered metric spaces by using monotone non-decreasing mapping.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.227-245
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• 2011

In this paper, we introduce a new iterative method for finding a common element of the set of solutions for mixed equilibrium problem, the set of solutions of the variational inequality for a ${\beta}$inverse-strongly monotone mapping and the set of fixed points of a family of finitely nonexpansive mappings in a real Hilbert space by using the viscosity and Ces$\`{a}$ro mean approximation method. We prove that the sequence converges strongly to a common element of the above three sets under some mind conditions. Our results improve and extend the corresponding results of Kumam and Katchang [A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mapping, Nonlinear Analysis: Hybrid Systems, 3(2009), 475-86], Peng and Yao [Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems, Mathematical and Computer Modelling, 49(2009), 1816-828], Shimizu and Takahashi [Strong convergence to common fixed points of families of nonexpansive mappings, Journal of Mathematical Analysis and Applications, 211(1) (1997), 71-83] and some authors.